The earliest traces of notated two-part music actually appear
from the end of the 10th century in central France, with theoretical
discussion surviving from the 9th century. The former are the
famous Chartres Manuscripts, the source of the present recording.
It was once thought that this music (organum) was not recoverable,
because it is not written with specified pitches. However when
the theoretical treatises are combined with the notation, it becomes
fairly straight-forward to learn the notes (according to Wulf Arlt
who was principally responsible for this reconstruction).

Of course, it is generally believed that composed polyphony grew
out of the practice of improvising a second line to a pre-existing
chant. What is unknown is how long it might have been going on.
A related practice which became codified at around this time was
troping, or adding text with a different melody into the middle of
a pre-existing chant. Tropes from this era have been similarly
reconstructed here, and indeed the practice of troping and of
singing a second melody seem to have been closely intertwined.
This is evident in the famous Winchester Troper (c.1000) from
England, not represented in the present program, which also includes
some of the earliest surviving written organum.

Although the surviving body of organa from this era is not large,
and what does survive appears to be almost an after-thought in
manuscript, there is actually a large degree of variety present. While
the end of the two-part section was always signified by returning to
the pitch of the underlying chant, and simple intervals were generally
employed, many different ideas were used within this general
framework. It has been speculated that only pieces which were
exceptional in some sense have been notated. At any rate, the later
repertory of St. Martial de Limoges &
Notre Dame, while larger & more complex,
are perhaps more uniform bodies of music.

A recording featuring the Winchester Troper:

Music for a King: The Winchester Troper

From Mss. of the XIth Century
Works by Pierre Chépélov & Joël Rust
Discantus - Brigitte LesneAeon 1436

And a recording presenting an excellent selection of songs (i.e.,
those pieces without an explicit liturgical context, although
devoted overwhelmingly to theological themes) highlighting early
polyphonic developments in France:

Returning more specifically to the subject of the Chartres
Manuscripts from which the present program is drawn, these are
written in what are called "staffless neumes" with
ligatures indicating the direction of pitch movement from one note
to the next. It is therefore a sort of relative pitch system.
The specification of notes as we know them did not really exist
before the landmark work of Guido d'Arezzo in his Micrologus
(c.1030). However earlier theoretical & pedagogical examples,
such as the famous Musica Enchiriadis (c.900) which uses a
simple alphabetical designation, used different combinations of
grid lines or letters. The Chartres Manuscripts can therefore be
seen as the earliest surviving practical written polyphonic examples,
together with the isolated contemporaneous Winchester Troper, and
as a real attempt to make something trained singers could use rather
than reading about in treatises. The later neumatic notation
follows more or less continuously from here, with the introduction
of gridlines and the concurrent sol-fa hexachords.